Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C22×C43 |
| direct product | C42×#108 |
| subgroup | G/H | |
|---|---|---|
| center | C43 | D4 |
| commutator | C4 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C43 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1470 conjugacy classes of subgroups; only the 906 normal subgroups are shown.
This group has 1470 conjugacy classes of subgroups — too many to draw. Only its 906 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4);(5,6,7,8);(9,10,13,16,14,17,12,22)(11,18,20,23,21,24,19,15);(9,11,12,19,14,21,13,20)(10,15,16,18,17,23,22,24)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260422);